3.813 \(\int \frac{a+\frac{b}{x^2}}{\left (c+\frac{d}{x^2}\right )^{3/2}} \, dx\)

Optimal. Leaf size=45 \[ \frac{a x}{c \sqrt{c+\frac{d}{x^2}}}-\frac{b c-2 a d}{c^2 x \sqrt{c+\frac{d}{x^2}}} \]

[Out]

-((b*c - 2*a*d)/(c^2*Sqrt[c + d/x^2]*x)) + (a*x)/(c*Sqrt[c + d/x^2])

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Rubi [A]  time = 0.103355, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{a x}{c \sqrt{c+\frac{d}{x^2}}}-\frac{b c-2 a d}{c^2 x \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)/(c + d/x^2)^(3/2),x]

[Out]

-((b*c - 2*a*d)/(c^2*Sqrt[c + d/x^2]*x)) + (a*x)/(c*Sqrt[c + d/x^2])

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Rubi in Sympy [A]  time = 10.7126, size = 37, normalized size = 0.82 \[ \frac{a x}{c \sqrt{c + \frac{d}{x^{2}}}} + \frac{2 a d - b c}{c^{2} x \sqrt{c + \frac{d}{x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)/(c+d/x**2)**(3/2),x)

[Out]

a*x/(c*sqrt(c + d/x**2)) + (2*a*d - b*c)/(c**2*x*sqrt(c + d/x**2))

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Mathematica [A]  time = 0.0339211, size = 33, normalized size = 0.73 \[ \frac{a c x^2+2 a d-b c}{c^2 x \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)/(c + d/x^2)^(3/2),x]

[Out]

(-(b*c) + 2*a*d + a*c*x^2)/(c^2*Sqrt[c + d/x^2]*x)

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Maple [A]  time = 0.008, size = 43, normalized size = 1. \[{\frac{ \left ( a{x}^{2}c+2\,ad-bc \right ) \left ( c{x}^{2}+d \right ) }{{x}^{3}{c}^{2}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)/(c+d/x^2)^(3/2),x)

[Out]

(a*c*x^2+2*a*d-b*c)*(c*x^2+d)/((c*x^2+d)/x^2)^(3/2)/x^3/c^2

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Maxima [A]  time = 1.423, size = 72, normalized size = 1.6 \[ a{\left (\frac{\sqrt{c + \frac{d}{x^{2}}} x}{c^{2}} + \frac{d}{\sqrt{c + \frac{d}{x^{2}}} c^{2} x}\right )} - \frac{b}{\sqrt{c + \frac{d}{x^{2}}} c x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/(c + d/x^2)^(3/2),x, algorithm="maxima")

[Out]

a*(sqrt(c + d/x^2)*x/c^2 + d/(sqrt(c + d/x^2)*c^2*x)) - b/(sqrt(c + d/x^2)*c*x)

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Fricas [A]  time = 0.232242, size = 63, normalized size = 1.4 \[ \frac{{\left (a c x^{3} -{\left (b c - 2 \, a d\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{c^{3} x^{2} + c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/(c + d/x^2)^(3/2),x, algorithm="fricas")

[Out]

(a*c*x^3 - (b*c - 2*a*d)*x)*sqrt((c*x^2 + d)/x^2)/(c^3*x^2 + c^2*d)

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Sympy [A]  time = 10.101, size = 65, normalized size = 1.44 \[ a \left (\frac{x^{2}}{c \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{2 \sqrt{d}}{c^{2} \sqrt{\frac{c x^{2}}{d} + 1}}\right ) - \frac{b}{c \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)/(c+d/x**2)**(3/2),x)

[Out]

a*(x**2/(c*sqrt(d)*sqrt(c*x**2/d + 1)) + 2*sqrt(d)/(c**2*sqrt(c*x**2/d + 1))) -
b/(c*sqrt(d)*sqrt(c*x**2/d + 1))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{a + \frac{b}{x^{2}}}{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/(c + d/x^2)^(3/2),x, algorithm="giac")

[Out]

integrate((a + b/x^2)/(c + d/x^2)^(3/2), x)